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## Phase detector

The goal of the PLL is to maintain a demodulating sine and cosine that match the incoming carrier. Suppose ${\omega }_{c}$ is the believed digital carrier frequency. We can then represent the actual received carrier frequency as theexpected carrier frequency with some offset, $\stackrel{˜}{{\omega }_{c}}={\omega }_{c}+\stackrel{˜}{\theta }(n)$ . The NCO generates the demodulating sine and cosine with the expected digital frequency ${\omega }_{c}$ and offsets this frequency with the output of the loop filter. The NCO frequency can then be modeled as $\stackrel{^}{{\omega }_{c}}={\omega }_{c}+\stackrel{^}{\theta }(n)$ . Using the appropriate trigonometric identities $\cos A\cos B=1/2(\cos (A-B)+\cos (A+B))$ and $\cos A\sin B=1/2(\sin (B-A)+\sin (A+B))$ . , the in-phase and quadrature signals can be expressed as

${z}_{0}(n)=1/2(\cos (\stackrel{˜}{\theta }(n)-\stackrel{^}{\theta }(n))+\cos (2{\omega }_{c}+\stackrel{˜}{\theta }(n)+\stackrel{^}{\theta }(n)))$
${z}_{Q}(n)=1/2(\sin (\stackrel{˜}{\theta }(n)-\stackrel{^}{\theta }(n))+\sin (2{\omega }_{c}+\stackrel{˜}{\theta }(n)+\stackrel{^}{\theta }(n)))$
After applying a low-pass filter to remove the double frequency terms, we have
${y}_{1}(n)=1/2\cos (\stackrel{˜}{\theta }(n)-\stackrel{^}{\theta }(n))$
${y}_{Q}(n)=1/2\sin (\stackrel{˜}{\theta }(n)-\stackrel{^}{\theta }(n))$
Note that the quadrature signal, ${z}_{Q}(n)$ , is zero when the received carrier and internallygenerated waves are exactly matched in frequency and phase. When the phases are only slightly mismatched we can use therelation
$\forall \theta , \mathrm{small}\colon \sin \theta \approx \theta$
and let the current value of the quadrature channel approximate the phase difference: ${z}_{Q}(n)\approx \stackrel{˜}{\theta }(n)-\stackrel{^}{\theta }(n)$ . With the exception of the sign error, this difference is essentially how much we need to offset our NCOfrequency If $\stackrel{˜}{\theta }(n)-\stackrel{^}{\theta }(n)> 0$ then $\stackrel{^}{\theta }(n)$ is too large and we want to decrease our NCO phase. . To make sure that the sign of the phase estimate is right, in this example the phase detector issimply negative one times the value of the quadrature signal. In a more advanced receiver, information from boththe in-phase and quadrature branches is used to generate an estimate of the phase error. What should the relationship between the I and Q branches be fora digital QPSK signal?

## Loop filter

The estimated phase mismatch estimate is fed to the NCO via a loop filter, often a simple low-pass filter. For thisexercise you can use a one-tap IIR filter,

$y(n)=\beta x(n)+\alpha y(n-1)$
To ensure unity gain at DC, we select $\beta =1-\alpha$

It is suggested that you start by choosing $\alpha =0.6$ and $K=0.15$ for the loop gain. Once you have a working system, investigate the effects of modifying these values.

## Matlab simulation

Simulate the PLL system shown in [link] using MATLAB. As with the DLL simulation, you will have to simulate the PLL on a sample-by-sample basis.

Use [link] to implement your NCO in MATLAB. However, to ensure that the phase term does not grow toinfinity, you should use addition modulo $2\pi$ in the phase update relation. This can be done by setting $\theta (n)=\theta (n)-2\pi$ whenever $\theta (n)> 2\pi$ .

[link] illustrates how the proposed PLL will behave when given a modulated BPSK waveform. In this case thetransmitted carrier frequency was set to $\stackrel{˜}{{\omega }_{c}}=\frac{\pi }{2}+\frac{\pi }{1024}$ to simulate a frequency offset.

Note that an amplitude transition in the BPSK waveform is equivalent to a phase shift of the carrier by $\frac{\pi }{2}$ . Immediately after this phase change occurs, the PLL begins to adjust the phase to force the quadraturecomponent to zero (and the in-phase component to $1/2$ ). Why would this phase detector not work in a real BPSK environment? How could it be changed to work?

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Cied
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At high concentrations (>0.01 M), the relation between absorptivity coefficient and absorbance is no longer linear. This is due to the electrostatic interactions between the quantum dots in close proximity. If the concentration of the solution is high, another effect that is seen is the scattering of light from the large number of quantum dots. This assumption only works at low concentrations of the analyte. Presence of stray light.
the Beer law works very well for dilute solutions but fails for very high concentrations. why?
how did you get the value of 2000N.What calculations are needed to arrive at it
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